Brushless permanent magnet machine with reduced cogging and torque ripple and method of producing the same

ABSTRACT

An electrical machine comprises a stator and a rotor. The stator includes a plurality of stator teeth and a winding with compact coils disposed on the teeth. The winding has (m) phases and (p) poles; where (m) is an integer and is greater than or equal to two; where (p) is an integer, is greater than or equal to two, and is an even number. The rotor has a magnetization pattern can include a magnetization skew. In some constructions, each stator includes one or more channels along a surface adjacent to the rotor. The channels can be of various shapes including a trapezoidal shape or a curvilinear shape.

RELATED APPLICATIONS

This application is a continuation patent application of U.S. patentapplication Ser. No. 10/625,926, filed on Jul. 24, 2003 now U.S. Pat.No. 6,940,198, titled BRUSHLESS PERMANENT MAGNET MACHINE WITH REDUCEDCOGGING AND TORQUE RIPPLE AND METHOD OF PRODUCING THE SAME, the entirecontent of which is incorporated herein by reference.

FIELD OF THE INVENTION

The invention relates to an electrical brushless permanent magnetmachine with reduced cogging and ripple torque.

BACKGROUND

Electrical machines, such as brushless permanent magnet (BLPM) motors,typically encounter problems with cogging and ripple torque, both ofwhich cause noise and vibration and can negatively affect the motorstarting performance. The sum of the cogging and ripple torquecomponents is defined as the electrical machine pulsating torque. Rippletorque is characterized as a cyclical variation in a delivered torque toa load caused by the interaction of the rotor magnetic field withharmonics in the stator current magnetomotive forces (mmf's). Coggingtorque describes the non-uniform torque as a result of the interactionof the rotor magnetization and angular variations in an air gappermeance (or reluctance) associated with the shape of the slots of thestator. By definition, no stator excitation is involved in coggingtorque production. There is a demand for an electric motor thatminimizes the effect of cogging and ripple torque and that exhibitssmooth operation. Further, there is a demand for an electrical machinehaving a rotor operable to provide a plurality of power or motor ratingsfor a given motor frame, and thereby reduce tooling costs and inventory.Further, there is a demand for an electrical machine to be easilyconfigurable for operation with different combinations of the number ofphases and poles.

SUMMARY

In one embodiment, the invention provides an electrical machine having amachine output rating. The electrical machine includes a shaft rotatableabout an axis, a rotor mounted or coupled to the shaft and rotating withthe shaft, and a stator including a stator core and windings. The rotoris configurable to include a first rotor portion having a relation to afirst output rating and a second rotor portion having a relation to asecond output rating. The stator core is configurable to be disposedadjacent to the first rotor portion when the machine output ratingcorresponds to the first output rating and adjacent to the second rotorportion when the machine output rating corresponds to the second outputrating.

In another embodiment, the invention provides an electrical machine thatcan be set up for operation in one of a plurality of modes including afirst mode wherein the electrical machine includes a first machineoutput rating and a second mode where the electrical machine includes asecond machine output rating. The second machine output rating isdifferent than the first machine output rating. The electrical machineincludes a shaft rotatable about an electrical machine axis, a rotormounted or coupled to the shaft and rotating with the shaft, and astator including a stator core and windings. The rotor is a first rotorin the first mode and a second rotor in the second mode. The first rotorhas a first rotor length and/or a first magnetization patterncorresponding to the first machine output rating. The second rotor has asecond rotor length and/or a second magnetization pattern correspondingto the second output rating. The stator core is a first core in thefirst mode and a second core in the second mode. The first core has afirst core length corresponding to the first machine output rating, andthe second core has a second core length corresponding to the secondmachine output rating.

In yet another embodiment, the invention provides an electrical machinehaving a shaft rotatable about an electrical machine axis, a rotormounted or coupled to the shaft and rotating with the shaft, and astator including a plurality of stator teeth. Each stator tooth includesone or more channels along a surface adjacent to the rotor. The channelincludes one of a trapezoidal shape, and a curvilinear shape.

In another embodiment, the invention provides a method of manufacturingan electrical machine having a stator and a rotor. The method includesthe acts of determining a desired output rating from a plurality ofoutput ratings; determining a length of the stator, the length having arelation to the desired output rating; determining a length of therotor, the length having a relation to the desired output rating;producing the stator; providing a magnetizer configured to magnetize therotor into a plurality of sections; and producing the rotor. The act ofproducing the rotor includes magnetizing the rotor to include a firstsection when the desired output rating corresponds to the first outputrating and magnetizing the rotor to include the first section and asecond section when the desired output rating corresponds to the secondoutput rating.

Other aspects of the invention will become apparent by consideration ofthe detailed description and accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is partial exploded view of the stator and rotor of a brushlesspermanent magnet electrical machine.

FIG. 2 is a longitudinal view of one construction of the rotor of FIG.1.

FIG. 3 is a longitudinal view of another construction of the rotor ofFIG. 1.

FIG. 4 is a longitudinal view of yet another construction of the rotorof FIG. 1.

FIG. 5 is a longitudinal view of another construction of the rotor ofFIG. 1.

FIG. 6 is a cross-sectional view of a stator core and a rotor capable ofbeing used in the electrical machine of FIG. 1.

FIG. 7 is a partial cross-sectional view of a portion of a stator corecapable of being used in the electrical machine of FIG. 1.

FIG. 8 is a partial cross-sectional view of a portion of a stator corecapable of being used in the electrical machine of FIG. 1.

FIG. 9 is a partial cross-sectional view of a portion of a stator corecapable of being used in the electrical machine of FIG. 1.

FIG. 10 a is a combination longitudinal view of a rotor andlongitudinal-sectional view of a stator, which has the same core lengthas the rotor.

FIG. 10 b is a combination longitudinal view of a rotor andlongitudinal-sectional sectional view of a stator, which has a shortercore length than the rotor and spacers are used to axially align thestator and the rotor.

FIG. 11 is an example of a stator-winding pattern in a double-layerarrangement with compact coils for an 18-slot, 12-pole, 3-phase machine.

FIG. 12 is an example of a stator-winding pattern in a single-layerarrangement with compact coils for an 18-slot, 12-pole, 3-phase machine.

FIG. 13 is an isometric view showing the geometry used to define an arcof magnetization skew (β) on the rotor.

DETAILED DESCRIPTION

Before any embodiments of the invention are explained in detail, it isto be understood that the invention is not limited in its application tothe details of construction and the arrangement of components set forthin the following description or illustrated in the following drawings.The invention is capable of other embodiments and of being practiced orof being carried out in various ways. Also, it is to be understood thatthe phraseology and terminology used herein is for the purpose ofdescription and should not be regarded as limiting. The use of“including,” “comprising,” or “having” and variations thereof herein ismeant to encompass the items listed thereafter and equivalents thereofas well as additional items. The terms “connected,” “coupled,” and“mounted” and variations thereof herein are used broadly and, unlessotherwise stated, encompass both direct and indirect connections,couplings, and mountings. In addition, the terms connected and coupledand variations thereof herein are not restricted to physical andmechanical connections or couplings.

FIG. 1 is a partial exploded view of the stator and rotor of oneconstruction of an electrical machine (e.g., motor, generator, etc.).For FIG. 1, the electrical machine is a motor 10 having a rotor 15 and astator 20. The rotor 15 is coupled to a shaft 17. In general, the stator20 receives electrical power, and produces a magnetic field in responsethereto. The magnetic field of the stator 20 interacts with a magneticfield of the rotor 15 to produce mechanical power on the shaft 17. Theinvention below refers to the electrical motor 10, however the inventionis not limited to the motor 10.

The rotor 15 includes a plurality of magnetic poles 25 of alternatingpolarity exhibited on a surface of a rotor core 30. The rotor core 30includes laminations (e.g., magnetic steel laminations), and/or solidmaterial (e.g., a solid magnetic steel core), and/or compressed powderedmaterial (e.g., compressed powder of magnetic steel). One constructionof the rotor 15 includes a sheet of permanent magnet (e.g., hardmagnetic) material disposed on the rotor core 30. Another constructionof the rotor 15 can include a plurality of strips of permanent magnetmaterial attached (e.g., with adhesive) around the core 30. Thepermanent magnet material can be magnetized by a magnetizer to provide aplurality of alternating magnetic poles. Additionally, the number ofmagnetic strips can be different than the number of rotor magneticpoles. Yet another construction of the rotor 15 contains blocks ofpermanent magnet material placed inside the rotor core 30.

The description of the invention is not limited to a particularmechanical construction, geometry, or position of the rotor 15. Forexample, FIG. 1 shows the rotor 15 located inside and separated by aradial air gap from the stator 20. In another construction, the rotor 15can be positioned radially exterior to the stator 20 (i.e., the machineis an external- or outer-rotor machine.)

One method to reduce cogging and ripple torque is skewing themagnetization of the magnetic poles 25 with respect to the stator 20.Alternatively, stator teeth of the stator 20 can be skewed with respectto the rotor magnetization. The optimal arc of skew in the magnetizationof the rotor is dependent on the electrical machine topology andparticular machine design. As shown in FIGS. 1–5, the “magnetization” ofthe rotor 15 refers to the line pattern 31 along the length of the rotor15 delineating alternating magnetic poles 25 on the rotor core 30. Eventhough a rotor 15 of the invention can include any number of alternatingmagnetic poles 25, FIGS. 2–5 show only one line pattern along the rotor15 for the sake of simplicity.

FIG. 13 illustrates the geometrical concepts involved in defining themagnetization skew of the rotor. The arc of magnetization skew can bedefined as the arc (β), measured in radians in between the longitudinallines 32 and 33 (see FIG. 2) on the rotor surface facing the air-gap,which separates the stator and the rotor.

FIG. 2 is a schematic diagram of one construction of the rotor 15divided into a plurality of axial sections 55 (e.g., 70, 71, and 72)along the rotational axis 50 of the rotor 15. The number of axialsections 55 can vary and is not limiting on the invention. An axialsection 55 refers to a portion of the rotor 15 differentiated byimaginary lines 60. Imaginary lines 60 refer to locations on the rotor15 where the direction of skew of the magnetization pattern 31 changes.One construction of the rotor 15 includes alternating magnetic poleswith substantially the same arc of magnetization skew (β) along eachaxial section 55, resulting in a herringbone pattern of magnetization.The length of each axial section 55 can vary. The arc of magnetizationskew is generally the same for each axial section 55 in order to ensurethe continuity of the magnetic poles, and is selected such as tominimize cogging and ripple torque. However, the outer axial sections(denoted by 95 and 100 in FIG. 5) can have a different arc ofmagnetization skew as it will be explained later.

The design (e.g., the length and magnetization pattern) of the rotorrelates to the desired output rating (e.g., power rating in horsepoweror torque and speed rating) of the electrical machine, where the desiredpower output rating is one of multiple ratings for the profile of thestator and rotor. Herein, the cross-sectional profile of the stator androtor refers to the cross-sectional geometry of the stator core 105 androtor 15. For example, FIG. 6 shows the profile of one construction ofthe stator core 105 and rotor 15 of the invention. The electricalmachine 10, in one construction, provides a plurality of output ratingsusing the same profile of the stator core 105 by varying the length(e.g., the number of laminations in the stack) of the stator core inmagnetic interaction with a respective combination of axial sections 55of the rotor 15. This aspect of the invention reduces tooling costs andinventory. In some constructions, when varying by design the statorstack length, the winding pattern is kept the same and the number ofturns and wire size are changed in order to match the electrical powersupply conditions, the desired output rating, and other designrequirements, such as the copper fill factor.

One method of providing the herringbone magnetization pattern (seeFIG. 1) on the rotor 15 includes the use of a magnetizer and amagnetizing fixture. Normally, a particular magnetizing fixture isrequired for an electrical machine having a particular length of rotor15. The rotor 15 of the invention allows the same magnetizer andmagnetizing fixture to be used for multiple output power ratings,thereby reducing tooling costs.

FIG. 2 shows one construction of the rotor 15 including three axialsections 70, 71, and 72. The stator 20 interacts with one or more of thethree axial sections 70, 71, and 72 to provide multiple output ratingsfor the profile of the motor. The first axial section 70 includesmagnetic poles aligned with a first skew direction, the second axialsection 71 includes magnetic poles aligned with a second skew direction,and the third axial section 72 includes magnetic poles aligned with thefirst skew direction. The second axial section 71 interacts with a firststator 20 to provide a minimum rating for the profile of the motor(e.g., a one-half horsepower output). A combination of the first 70 andsecond 71 axial sections interact with a second stator 20 to provide anintermediate rating for the profile of the motor (e.g., a three-quarterhorsepower output). A combination of all three axial sections 70, 71,and 72 interact with a third stator 20 to provide a maximum rating forthe profile of the motor (e.g., a one horsepower output). Thisconstruction of the rotor 15 includes fewer changes in skew directionalong the rotor with respect to the description of the otherconstructions given below, such that the magnetizing fixture includes amore simple and accurate magnetization pattern. Making the first 70 andthird 72 axial sections of the same length and arc of magnetization skew(β), contributes to the axial magnetic symmetry of the motor having themaximum rating for the profile of the motor. However, with aconstruction of the rotor 15 as shown in FIG. 2, characterized asincluding a number of axial sections equal to the number of possibleoutput ratings within the profile of the motor 10, the axial symmetry ofa motor of an intermediate output rating is not necessarily guaranteed.

FIG. 3 is a schematic diagram of yet another construction of the rotor15. The construction shown in FIG. 3 is characterized as including anumber of axial sections equal to double the number of possible outputratings within the profile of the motor 10. For example and as shown inFIG. 3, the rotor 15 includes six axial sections 75, 76, 77, 78, 79, and80 operable to provide three power output ratings. The third 77 andfourth 78 axial sections interact with a first stator 20 to provide aminimum rating for the profile of the motor (e.g., one-half horsepoweroutput). A combination of the second 76, third 77, fourth 78, and fifth79 axial sections interact with a second stator 20 to provide anintermediate rating for the profile of the motor (e.g., a three-quarterhorsepower output). All six axial sections 75, 76, 77, 78, 79, and 80interact with a third stator 20 to provide a maximum rating for theprofile of the motor (e.g., a one horsepower output). The first, second,and third stators 20 described above include the same profile of statorcore 105, but can differ in the length (e.g., number of laminations) ofthe stator core to provide the desired output rating.

FIG. 4 is a schematic diagram of yet another construction of the rotor15 of the invention. This construction is characterized by the number ofaxial sections equal to double minus one the number of possible outputratings within the profile of the motor 10. For example, FIG. 4 showsthe rotor 15 including five axial sections 85, 86, 87, 88, and 89operable to provide three power output ratings. The third 87 axialsection interacts with a first stator 20 to provide a minimum rating forthe profile of the motor (e.g., one-half horsepower output). Acombination of the second 86, third 87, and fourth 88 axial sectionsinteract with a second stator 20 to provide an intermediate rating forthe profile of the motor (e.g., a three-quarter horsepower poweroutput). A combination of the all five axial sections 85, 86, 87, 88,and 89 interact with a third stator 20 to provide a provide a maximumrating for the profile of the motor (e.g., a one horsepower output).Each of the sections 85, 86, 87, 88, and 89 can be of different length.The first, second, and third stators 20 described above include the sameprofile of stator core 105, but can differ in the length (e.g., numberof laminations) of the stator core to provide the desired output rating.

The constructions of the invention shown in FIG. 3 and FIG. 4 providemore freedom to vary by design the lengths of the axial sections andimprove the axial symmetry of the motor. In the rotor construction shownin FIG. 3, the axial symmetry of the motor is improved if the first 75and sixth 80, the second 76 and the fifth 79, the third 77 and thefourth 78 axial sections, have, respectively, equal length and arc ofmagnetization skew. In the rotor construction shown in FIG. 4 the axialsymmetry of the motor is improved if the first 85 and the fifth 89, thesecond 86 and the fourth 88 axial sections have, respectively, equallength and arc of magnetization skew. In the rotor construction shown inFIG. 5 the axial symmetry of the motor is improved if the first 95 andthe sixth 100, the second 96 and the fifth 99 axial sections, and thethird 97 and the fourth 98 axial sections have, respectively, equallength and arc of magnetization skew.

One construction of a rotor 15 design includes a first one or more axialsections in relation to a first output rating (P₁) (e.g., one-halfhorsepower output). The first one or more axial sections 55 have a firstlength (L₁). A ratio of the first length L₁ of the first one or moreaxial sections 55 divided by a maximum length (L_(m)) of the rotor, usedfor a maximum rating (P_(m)) for the profile of the motor, is in a rangeof ¾ to 1½ times the ratio of the power ratings (P₁/P_(m)), with apreferred range of ¾ to 1¼ times the ratio of the power ratings(P₁/P_(m)). This range of power and length ratio provides the designerwith freedom to design for a desired output rating by trading off, onone hand the motor size and cost, and on the other hand motorefficiency. The rotor 15 can also include a second one or more axialsections in relation to an intermediate rating for the profile of themotor (P₁) (e.g., a half horsepower output). The second one or moreaxial sections have a second length (L_(i)), and the second lengthincludes the first length. A ratio of the second length of the secondone or more axial sections divided by a maximum length of the rotor(L_(m)) is in a range of ¾ to 1½ of the ratio of the power ratings(P_(i)/P_(m)), with a preferred range of ¾ to 1¼ times the ratio of thepower ratings (P_(i)/P_(m)).

The total number of axial sections and the total number of ratings for agiven motor profile are not limiting on the invention. Therefore,generally speaking, a rotor 15 design includes one or more axialsections in relation to a first output rating (P_(x)) (e.g., one-halfhorsepower output, ¾ horsepower output, etc.). The one or more axialsections 55 have a first total length (L_(x)). A ratio of the firsttotal length L_(x) of the one or more axial sections 55 divided by amaximum length (L_(m)) of the rotor, used for a maximum rating (P_(m))for the profile of the motor, is in a range of ¾ to 1½ times the ratioof the power ratings (P_(x)/P_(m)), with a preferred range of ¾ to 1¼times the ratio of the power ratings (P_(x)/P_(m)).

In addition to reducing cogging and ripple torque, the arc ofmagnetization skew also affects the specific torque output (e.g., torqueper unit axial length at a given current) of the motor 10. In general,the torque output or power rating decreases as the arc of magnetizationskew increases. Reducing the arc of magnetization skew can increase themotor torque output per axial length. Accordingly, at the penalty ofincreasing the cogging and ripple torque, reducing the arc ofmagnetization skew allows shortening of the axial length of the rotor 15and maintaining a desired power output of the motor 10. Shortening theaxial length of rotor 15 reduces material costs.

FIG. 5 shows yet another construction of the rotor 15 including innersections 96, 97, 98, and 99 having substantially equal arc ofmagnetization skew, and outer sections 95 and 100 having a lesser arc ofmagnetization skew with respect to the magnetization skew of the innersections 96, 97, 98, 99. The inner sections 96, 97, 98, 99 include thesame arc of magnetization skew to enhance continuity and symmetry. Theouter sections 95 and 100 have a lesser arc of magnetization skew (β) toenhance the output rating of the electrical machine with use of all sixaxial sections 95, 96, 97, 98, 99, and 100.

For FIGS. 2–5 described above, the value and number of output ratingscan vary and is not limiting on the invention. In addition, theincremental difference between output ratings related to one or acombination of axial sections 55 can vary and is not limiting on theinvention.

In one construction of the invention, the electrical machine includesthe rotor 15 having a plurality of axial sections as shown in one ofFIGS. 2–5. The rotor 15 provides multiple output ratings for the profileof the machine. This construction of the electrical machine uses thesame rotor 15 in electrical machines of varying output ratings, therebyreducing the part count and the inventory required for producing a rangeof motors of different output ratings using the same motor profile.

In another construction of the electrical machine, one or more axialsections 55 of the rotor 15 are not present when the desired outputrating of the electrical machine is less than the maximum power outputrating for the machine to be produced using the same stator coreprofile. For example, for the electrical machine having a desired outputrating of one-half horsepower, the rotor axial sections 70 and 72 ofFIG. 2 are not necessary. This construction of the electrical machineallows the use of the same magnetizer to magnetize the rotor 15 having arange of output ratings within the profile of the machine. In addition,this construction reduces the material waste (e.g., 70 and 72 of FIG. 2)of the rotor 15.

Various designs of stator 20 can be used to interact with eachconstruction of the rotor 15 described above and shown in FIGS. 2–5. Thefollowing is a description of one construction of the invention thatincludes the rotor 15 disposed radially from the stator 20. Withreference to FIG. 1, the stator 20 includes a stator core 105 having aplurality of stator teeth 110 and stator windings 112. In oneconstruction, the stator core 105 includes a stack of magnetic steellaminations or sheets. In other constructions, the stator core 105 isformed from a solid block of magnetic material, such as compacted powderof magnetic steel. The stator windings 112 include electrical conductorsplaced in the slots 120 (FIG. 6) and around the plurality of teeth 110.Other constructions and types of the stator core 105 and stator windings112 known to those skilled in the art can be used and are not limitingon the invention.

Electrical current flowing through the stator windings 112 produces amagnetic field that interacts with the magnetization of the rotor 15 toprovide torque to the rotor 15 and shaft 17. The electrical current canbe an (m) phase alternating current (AC), where (m) is an integergreater than or equal to two. The electrical current can have varioustypes of waveforms (e.g., square wave, quasi-sine wave, etc). The statorwindings 112 receive electrical current from an electrical drive circuit(not shown). One construction of the electrical drive circuit includes acontroller and an inverter with one or more power electronic switches(e.g., MOSFET, IGBT) to vary the flow of electrical current to thewindings dependent on various electrical machine operating parameters(e.g., speed, load, rotor position, etc.). To determine the position ofthe rotor 15, the control circuit includes, in some constructions, asensor (e.g., Hall effect device, encoder, etc.) to provide the controlcircuit with a signal representative of the rotor position.Alternatively, the control circuit can estimate the rotor positionthrough what is commonly referred as a sensorless control. Theelectrical drive circuit can include other components and circuitconstructions known to those skilled in the art and is not limiting onthe invention.

FIG. 6 shows a cross-sectional profile of a motor cross-sectionperpendicular to axis 50 used in one motor construction (the statorwindings 112 are not shown in FIG. 6). The stator core 105 includes theplurality of stator teeth 110, slots 120, and a back iron portion 115.Each of the plurality of stator slots 120 receives one or more statorcoils, the assembly of which constitutes the stator windings 112. Thestator windings receive a multi-phase electrical current, where thenumber of phases (m) is an integer greater than or equal to two. Thenumber (t) of stator teeth 110 equals the number of slots 120, where (t)is an integer. A slot 120 is defined by the space between adjacentstator teeth 110. The rotor 15 is produced, in one construction, byfixing three arc shaped magnets 26 on a rotor core 30. Other rotordesigns and constructions are also possible as mentioned previously. Amagnetizer is used to produce on the rotor 15 a number (p) ofalternating magnetic poles that interact with the stator 20, where (p)is an even (i.e., divisible by 2) integer greater than or equal to two.The stator core 105 includes a ratio of the number of stator teeth tomagnetic poles (t/p) equal to (m/2) or (m/4).

The stator core 105 having the above-described construction (see FIG. 6)can be used to design and manufacture motors with various (m) electricphases, with windings 112 composed of compact coils (see the windingpatterns in FIG. 11 and FIG. 12) and rotors having poles (p). Forexample, a stator core 105 having a same cross sectional profile with anumber (t) of stator teeth 110 can be used, in principle, to producemotors with (m) phases or an increased number of phases (km). In orderto maintain the same (t/p) ratio, the number of poles can be reduced to(p/k), and therefore (k) can be any integer for which (p/k) is an eveninteger greater than or equal to two. Alternatively, the number ofphases can be decreased from (m) to an integer (m/k), where (k) is anyinteger for which (m/k) is an integer greater than or equal to two. Inorder to maintain the same (t/p) ratio, the number of magnetic poles canbe increased to (kp).

To provide a stator 20 with (m) symmetrical electric phases, within eachphase the compact coils, belonging to the phase, are connected such thatconsecutive phases are placed at a mechanical angle of (4π/(mp))radians. For any number of phases (m), the number (t) of teeth andnumber (p) of poles is designed so that their ratio (t/p) is equal to(m/2) or (m/4). The number of teeth per pole and phase (t/p/m) istherefore a design constant, equal to ½ or ¼ respectively, and thereforefor constant air-gap magnetic loading (i.e. flux density), the magneticflux per tooth remains constant. Therefore, the stator teeth 110 can beoptimally designed for any number of phases.

In some constructions of the machine, it is generally desired for theback iron portion 115 (see FIG. 6) to operate at approximately the samemagnetic loading as the teeth 110. To equalize the magnetic loading, aminimum width of the back iron portion 115 can equal half the value ofthe product of the number of teeth per pole times the tooth width. Theminimum width (w_(y)) of the back iron is defined as the minimumdistance between the top of a slot 120 and a circle with the center onthe rotational axis 50 and a radius equal to the minimum distancebetween the rotational axis 50 and any of the flat surfaces from theoutside surface of the stator core 105 (see FIG. 6). The number of teethper pole (t/p) can be an integer or a fractional number. Limits on theminimum width of the back iron portion 115 include manufacturability andincreased mmf drop and core losses related to back-iron flux density.For a design with an increased number of phases (m), the number of poles(p) is decreased by design in order to maintain a constant ratio (t/p)for a given lamination. Lowering by design the number of poles (p)results in an increased magnetic pole pitch and, for the same air-gapmagnetic loading, an increased magnetic flux density in the back ironportion 115. Lowering by design the number of poles (p), also results ina decrease of the fundamental frequency of the magnetic field for agiven rotational speed of electrical machine and limits core losses inthe back-iron portion 115. Finite element analysis that considers theabove-described parameters indicates the width of the back iron portion115 ranges between (1½–4½) times the product of the number of teeth perpole (t/p) divided by 2 and times the tooth width (w_(t)).

One construction of the stator windings 112 includes a double layerarrangement of compact coils (FIG. 11), which are placed around eachtooth (i.e. the coils have a pitch of 1-slot). In this double layerarrangement, each slot is shared by two coil sides, each of the coilsides belonging to a different coil and phase. The two coil sidessharing a slot can be placed side by side or one on top of the other.The double-layer winding pattern for an example 18-slot, 12-pole,3-phase winding is shown in FIG. 11. Following the rules set above, fora given stator core and a winding with compact coils and a double layerpattern, the coil connections, the number of turns per coil and the wiresize can be modified by design in order for the machine to operate withany number of phases (m) and poles (p) for which (t/p) is equal to (m/2)or (m/4).

Another construction of the windings 112 includes a single layerarrangement of compact coils (FIG. 12), which are placed around everyother tooth (i.e. the coils have a pitch of 1-slot and are only placedaround half the number of teeth). In this single layer arrangement, eachslot contains only one coil side. The single layer winding pattern foran example 18-slot, 12-pole, 3-phase winding is shown in FIG. 12.Following the rules set above, for a given stator core and a windingwith compact coils and a single layer pattern, the coil connections, thenumber of turns per coil and the wire size can be modified by design inorder for the machine to operate with any number of phases (m) and poles(p) for which (t/p) is equal to (m/2) or (m/4). In comparison with adouble layer winding with compact coils, a single layer winding withcompact coils has only half the number of coils but the per phaseend-winding is generally longer.

The phase windings of the stator 20 are symmetrically and equidistantlydistributed at an angle of (2π/m) electrical radians or (4π/(mp))mechanical radians. A symmetrical (m) phase system of currents flowingthrough the stator windings produces a magnetomotive force (mmf) with aspace electrical fundamental harmonic of the mechanical order (p/2). Themmf also includes space harmonics of the electrical order (2km−1) and(2km+1), where k is an integer larger or equal to one. When theelectrical machine couples to a load, the mmf harmonics cause rippletorque, an undesired machine characteristic described above. Theamplitude of the mmf harmonic increases as its harmonic order decreases.The amplitude of the lower-order mmf harmonics (2m−1) and (2m+1) can besignificant and their reduction ensures a smooth motor operation.

With simple compact windings, built according to the previousdescription, conventional means of reducing the mmf harmonics, such asshort-pitching the winding are not available. Instead, an optimalmagnetization skew is determined and implemented to reduce the mmfharmonics and the ripple torque, as well as the cogging torque.

The skew factor for a v-th electrical order mmf space harmonic is givenby the equation: (k_(sv)=4 sin(νpβ/4)/(νpβ)), where the arc ofmagnetization skew (β) is measured in radians on the rotor surfacefacing the air-gap (see FIG. 2). A harmonic is completely eliminated ifthe argument of the sine function satisfies the equation (νpβ/4=nπ),where (n) is an integer larger or equal to zero. For an mmf harmonic ofthe electrical order (ν=2km−1) the previous equation is equivalent to(β=4nπ(p(2km−1))) and for an mmf harmonic of the electrical order(ν=2km+1) the previous equation is equivalent to (β=4nπ(p(2km+1))). Forincreasing values of (n) and/or (k), both arrays (4nπ/(p(2km−1))) and(4nπ/(p(2km+1))) converge to (2π(pm)).

Therefore, to reduce both (2km−1) and (2km+1) orders of mmf spaceharmonics, one construction of the motor 10 includes the stator 20having a ratio of stator teeth 110 per magnetic pole (t/p) equal to(m/2) and the rotor 15 including an arc of magnetization skew(β=2π/(pm)) measured in radians on the rotor surface facing the air-gap.Another construction of the motor 10 includes the stator 20 having aratio of stator teeth 110 per pole (t/p) equal to (m/4) and the rotor 15including an arc of magnetization skew (β=2π/(pm)) measured in radianson the rotor surface facing the air-gap.

A typical manufacturing technique to provide a double layer statorwinding with compact coils includes use of a needle or gun winder. Asubstantially large opening of the stator slot 120 is beneficial towardsthe air-gap in order to allow the needle of the winder to be insertedinto the slot.

A typical manufacturing technique to provide a single layer statorwinding with compact coils includes use of an insertion winder. Asubstantially large opening of the stator slot 120 is required in orderto allow the conductors to be inserted into the slot. Other types andtechniques known to those in the art to provide the stator windings 112of the stator 20 can be used.

A relatively large opening of slot 120 increases the ease of insertionof the needle winder and of the conductors of the windings,respectively. An opening of the slot 120 suitably large to becost-effective for automatic winding manufacturing includes a rangegreater than ⅙th of a tooth pitch. Tooth pitch is the distance betweenadjacent centerlines 135 (see FIGS. 7–9) of teeth 110. The slots 120create a variation of the permeance of the air-gap between the rotor 15and the stator 20. The variation in air-gap permeance interacts with themagnetic field of the rotor 15 to cause cogging torque. As noted above,cogging torque is an undesired characteristic of electrical machines andits minimization, by reducing the variation of the air-gap permeance,while still maintaining a slot opening suitably large for volumemanufacturing technologies.

FIGS. 7, 8, and 9 show a construction of the stator core 105 thatincludes “dummy” channels 130 in the stator teeth 110. The dummychannels 130 reduce the amplitude and, for certain motor designs, canincrease the frequency of the cogging torque, as it will be shown in thefollowing. The shape and dimensions of each dummy channel 130 are variedby design to provide a more symmetrical variation of the cogging torqueversus rotor position. Constructions of the stator 20 include a suitablecore 105 having one or two dummy channels 130 per tooth 110. Of course,the stator 20 of the invention can include more dummy channels 130 andis not limiting on the invention.

The number of equivalent openings of slots 120 of the stator 20 includesthe number of slots 120 and the number of dummy channels 130 (see FIGS.7–9). By adding a number (d) of dummy channels 130 at the free ends ofeach tooth 110, where (d) is an integer greater than or equal to zero,the number of equivalent slot openings towards the air-gap increasesfrom the number (t) to ([d+1]t). The frequency of the cogging torque isequal to the least common multiple of the number ([d+1]t) of equivalentslot openings and the number of poles (p).

For (t/p=m/2), mathematical induction proves the following:

Phases Poles Cogging frequency (m) (p) dummy channels (d) (Hz) 2k 2j 1mp 2k+1 2j 0 or 1 mp 2k+1 2j 2 3mpwhere (k) and (j) are integers greater than or equal to one. In each ofthe above cases, the arc of magnetization skew (β) equal to (2π/(mp)),as measured in radians in between the longitudinal lines 32 and 33 onthe rotor surface facing the air-gap (see FIG. 13), causes a reductionof both the cogging torque and torque ripple.

For (t/p=m/4), mathematical induction proves the following:

Phases Poles Cogging frequency (m) (p) Dummy channels (d) (Hz) 2k+1 4j 0or 1 mp 2k+1 4j 2 3mpwhere (k) and (j) are integers greater than or equal to one. In each ofthe above cases, the arc of magnetization skew (β) is equal to(2π/(mp)), as measured in radians in between the longitudinal lines 32and 33 on the rotor surface facing the air-gap (see FIG. 13), causes areduction of both the cogging torque and torque ripple.

FIG. 7 shows one construction of a stator tooth 110 including a dummychannel 130 having a centerline located at the middle of the tooth 110and coinciding with the tooth centerline 135. In the construction fromFIG. 7, the dummy channel 130 is generally trapezoidal-shaped and ischaracterized by a top width of channel (w_(n)) a bottom width ofchannel (w_(b)), and the side angle (α). The width of the slot opening(w_(o)) is designed to a minimum value for which cost-effectivemanufacturing of the stator winding is achieved and the cogging torqueis low. The height of the slot opening (h_(o)), and the dimensions ofthe dummy channels (w_(n)), (w_(b)) and (α) are designed to optimize themachine from a magnetic and mechanical point of view.

Finite element analysis of the electromagnetic field indicates aconstruction of the channel 130 of FIG. 7, including a top width (w_(n))ranging from (0.5w_(o))≦(w_(n))≦(1.5 w_(o)), a bottom width (w_(b))ranging from (0.3w_(o))≦(w_(b))≦(1.2w_(o)) and the side angle (α)ranging from (30°)≦(α)≦(135°), controls the local level of magneticsaturation in the tooth tip, modifies the air-gap magnetic permeance,reduces the cogging torque, and improves the symmetry of the coggingtorque variation against rotor position. Therefore, the cogging torqueis substantially reduced in a motor which has, in addition, the rotormagnetization skewed with the optimal arc of skew (β) previouslydetermined.

FIG. 8 shows a second construction of a stator tooth 110 including twodummy channels 130. The dummy channels are located so that theircenterlines 137 are dividing the slot pitch, which is contained inbetween the centerlines 136 of two adjacent slots, in three intervals ofapproximately equal length. In the construction from FIG. 8, the dummychannel 130 is generally trapezoidal-shaped and is characterized by atop width of channel (w_(n)), a bottom width of channel (w_(b)), a sideangle (α), and yet another side angle (γ). The width of the slot opening(w_(o)) is designed to a minimum value for which cost-effectivemanufacturing of the stator winding is achieved and the cogging torqueis low. The height of the slot opening (h_(o)), and the dimensions ofthe dummy channels (w_(n)), (w_(b)), (α), and (γ) are designed tooptimize the machine from a magnetic and mechanical point of view.

Finite element analysis of the electromagnetic field indicates aconstruction of the channel 130 of FIG. 8, including a top width (w_(n))ranging from (0.5w_(o))≦(w_(n))≦(1.5w_(o)), a bottom width (w_(b))ranging from (0.3w_(o))≦(w_(b))≦(w_(o)) and the side angles (α) and (γ)ranging from (30°)≦(α)≦(90°) and (30°)≦(γ)≦(90°), controls the locallevel of magnetic saturation in the tooth tip, modifies the air-gapmagnetic permeance, reduces the cogging torque, and improves thesymmetry of the cogging torque variation against rotor position.Therefore, the cogging torque is substantially reduced in a motor whichhas, in addition, the rotor magnetization skewed in the optimal arc ofskew (β) previously determined. For a construction of a stator 30including two dummy channels 130 per teeth 110, space limitations canlimit the value of the side angles (α) and (γ) to be equal or belowninety degrees.

FIG. 9 shows another construction of a stator tooth having twocurvilinear shaped dummy channels 130. The dummy channels are located sothat their centerlines 137 are dividing the slot pitch, which iscontained in between the centerlines 136 of two adjacent slots, in threeintervals of approximately equal length. The opening of the dummychannels 130 towards the air-gap is equal to the opening (w_(o)) of theslot 120. The curvilinear shape follows that of an arc of the circlewith the center on the respective dummy channel centerline and adiameter larger or equal to ¾ of (w_(o)) and smaller or equal to 1½ of(w_(o)). This shape and dimensions of the dummy channels reduce thecogging torque and increase the durability of the punching die used formanufacturing stator laminations.

Having described constructions of the electrical machine, a method ofassembling one construction of the electrical machine will now bedescribed. It is envisioned that the method may be modified for otherconstructions. Furthermore, it is envisioned that not all of the actsbelow may be required, that some of the acts may be modified, or thatthe order of the acts may vary.

A designer provides the rotor 15 having the plurality of alternatingmagnetic poles. The rotor 15 is divided into a plurality of portionsalong the longitudinal axis 50. The plurality of portions can include afirst portion related to a first output rating (e.g., one-halfhorsepower), a second portion relating to a second output rating (e.g.,three-quarter horsepower), and a third portion relating to a thirdoutput rating (e.g., one horsepower).

Each of the portions is divided into one or more axial sections 55(e.g., axial sections 70, 71, and 72 in FIG. 2). Each axial section 55includes a respective arc of magnetization skew (β) of the alternatingmagnetic poles in relation to the first, second, and third outputratings of the electrical machine. The arc of magnetization skew (β) ismeasured on the rotor surface facing the air-gap, as shown in FIG. 2. Amagnetizer is used to provide the magnetization of the axial sections ofthe rotor. This method of constructing the rotor 15 allows a commonmagnetizer to be used to provide the magnetization of the rotor for aplurality of desired output ratings of the electrical machine, therebyreducing tooling costs. In one construction of the electrical machine,the end axial sections 55 that are not needed to provide the desiredoutput rating are not included with the rotor 15 and therefore also thematerial cost is reduced. In another construction, all axial sections 55(e.g., sections in relation and not in relation to the desired outputrating) of the rotor 15 can be retained in the assembly of the rotor 15.This second construction is advantageous if, for example, the cost ofthe rotor material from the end axial sections that are not necessarilyrequired in relation to the desired output rating is smaller than thecost savings achieved by maintaining an inventory with only a reducednumber of rotor dimensions.

Using a uniform profile of the stator 20, the designer determines thelength of the stator core 105 to interact with the rotor 15 to providethe desired output rating. For example, with a laminated construction ofthe stator core 105, the designer selects a stack length of laminationsof magnetic material to provide the desired output rating. The statorcore 105 is wound with windings 112 designed for the electrical supplyconditions, the stator core length, the rotor length, and the desiredmotor output. The manufacturing operator aligns the stator 20 with therotor 15, so that the axial centerline of the stator core 105 coincideswith the axial centerline of the rotor 15 and no side-pull axial forcesare exhibited due to stator-rotor misalignment (see FIG. 10 a).Referring to FIG. 10 b, if the rotor 15 includes other end axialsections not in relation to the desired output rating (e.g. 75 and 80),additional end axial spacers 150 can be added to help align the statorcore 105 with the rotor 15. For example, referring to FIGS. 3 and 10 b,if the desired output rating is three-quarter horsepower, the stator 20would be aligned with the axial sections 76, 77, 78, and 79. Two spacers150 are used to cover the axial sections 75 and 80, respectively, of therotor 15 not in relation to the desired three-quarter horsepowermachine. The end-spacers can enhance support of assembly of theelectrical machine in a uniform housing. The axial length of spacers 150can vary with the constructions of the rotor 15 and stator 20 describedabove.

Thus, the invention provides, among other things, an electrical machinewith reduced cogging and torque ripple. Various features and advantagesof the invention are set forth in the following claims.

1. An electrical machine comprising: a rotor; a stator adapted tomagnetically interact with the rotor, the stator comprising a pluralityof stator teeth, the plurality of stator teeth comprising a first statortooth comprising a first free end and first and second channels disposedin the first stator tooth at the first free end, the first and secondchannels having a substantially curvilinear shape, a second stator toothdisposed circumferentially adjacent to the first stator tooth, thesecond stator tooth comprising a second free end and third and fourthchannels disposed in the second stator tooth at the second free end, thethird and fourth channels having the substantially curvilinear shape,and a third stator tooth disposed circumferentially adjacent to thesecond stator tooth, the third stator tooth comprising a third free endand fifth and sixth channels disposed in the third stator tooth at thethird free end, the fifth and sixth channels having the substantiallycurvilinear shape; wherein each of the six channels comprises arespective opening having an opening width (w_(o)); wherein the statordefines a first slot opening between the first stator tooth and thesecond stator tooth, and a second slot opening between the second statortooth and the third stator tooth, each of the first and second slotopenings having the width (w_(o)); wherein each of the curvilinearshapes of the six channels has a respective centerline, and each of thecurvilinear shapes of the six channels follows an arc of a circle with acenter on the respective centerline; and wherein the first slot openinghas a first slot centerline disposed between the first stator tooth andthe second stator tooth, the second slot opening has a second slotcenterline disposed between the second stator tooth and the third statortooth, and the centerlines of the third and fourth channels divide aninterval between the first and second slot centerlines into three equalintervals.
 2. An electrical machine as set forth in claim 1 wherein thecircle has a diameter (d) that satisfies the relationship((w_(o))≦(d)<(1.5w_(o))).
 3. An electrical machine as set forth in claim1 wherein the plurality of stator teeth further comprises a fourthstator tooth, and wherein the fourth stator tooth comprises a free endand two curvilinear-shaped channels disposed in the free end.
 4. Anelectrical machine as set forth in claim 1 wherein the electricalmachine comprises a brushless, permanent magnet motor comprising therotor and stator.
 5. An electrical machine comprising: a rotor; a statoradapted to magnetically interact with the rotor, the stator comprising aplurality of stator teeth, the plurality of stator teeth comprising afirst stator tooth comprising a first free end and a firstnon-rectangular, substantially trapezoidal shaped channel disposed inthe first stator tooth at the first free end, and a second stator toothcomprising a second free end and a second non-rectangular, substantiallytrapezoidal shaped channel disposed in the first stator tooth at thefirst free end; wherein the stator defines a first slot opening betweenthe first stator tooth and the second stator tooth, the slot openinghaving a width (w_(o)); wherein the shape of each of the first andsecond non-rectangular, substantially trapezoidal shaped channelsincludes an opening adjacent to the rotor, the opening having an openingwidth (w_(b)), a base at a depth from the surface, the base having abase width (w_(n)), and a pair of sides at a side angle (α) with respectto the base; and wherein (w_(n)) satisfies the relationship(0.5w_(o))≦(w_(n))≦(1.5w_(o)), (w_(b)) satisfies the relationship(0.3w_(o))≦(w_(b))≦(1.2 w_(o)), and (w_(n)) does not equal (w_(b)). 6.An electrical machine as set forth in claim 5 wherein the plurality ofstator teeth further comprises a third stator tooth, and wherein thethird stator tooth comprises a free end and a non-rectangular,substantially trapezoidal shaped channel disposed in the free end.
 7. Anelectrical machine as set forth in claim 5 wherein the first statortooth further comprises a third non-rectangular, substantiallytrapezoidal shaped channel disposed in the first stator tooth at thefirst free end.
 8. An electrical machine as set forth in claim 5 whereinthe electrical machine comprises a brushless, permanent magnet motorcomprising the rotor and stator.
 9. An electrical machine comprising: arotor; a stator adapted to magnetically interact with the rotor, thestator comprising a plurality of stator teeth, the plurality of statorteeth comprising a first stator tooth comprising a first free end and afirst non-rectangular, substantially trapezoidal shaped channel disposedin the first stator tooth at the first free end, and a second statortooth comprising a second free end and a second non-rectangular,substantially trapezoidal shaped channel disposed in the first statortooth at the first free end; wherein the shape of each of the first andsecond non-rectangular, substantially trapezoidal shaped channelsincludes an opening adjacent to the rotor, the opening having an openingwidth (w_(b)), a base at a depth from the surface, the base having abase width (w_(n)), and a pair of sides at a side angle (α) with respectto the base; and wherein (α) satisfies the relationship (30°)≦(α)<(90°).10. An electrical machine as set forth in claim 9 wherein the pluralityof stator teeth further comprises a third stator tooth, and wherein thethird stator tooth comprises a free end and a non-rectangular,substantially trapezoidal shaped channel disposed in the free end. 11.An electrical machine as set forth in claim 9 wherein the electricalmachine comprises a brushless, permanent magnet motor comprising therotor and stator.
 12. An electrical machine comprising: a rotor; astator adapted to magnetically interact with the rotor, the statorcomprising a plurality of stator teeth, the plurality of stator teethcomprising a first stator tooth comprising a first free end and a firstnon-rectangular, substantially trapezoidal shaped channel disposed inthe first stator tooth at the first free end, and a second stator toothcomprising a second free end and a second non-rectangular, substantiallytrapezoidal shaped channel disposed in the first stator tooth at thefirst free end; wherein the shape of each of the first and secondnon-rectangular, substantially trapezoidal shaped channels includes anopening adjacent to the rotor, the opening having an opening width(w_(b)), a base at a depth from the surface, the base having a basewidth (w_(n)), and a pair of sides at a side angle (α) with respect tothe base; and wherein (α) satisfies the relationship (90°)<(α)≦(135°).13. An electrical machine as set forth in claim 12 wherein the pluralityof stator teeth further comprises a third stator tooth, and wherein thethird stator tooth comprises a free end and a non-rectangular,substantially trapezoidal shaped channel disposed in the free end. 14.An electrical machine as set forth in claim 12 wherein the electricalmachine comprises a brushless, permanent magnet motor comprising therotor and stator.
 15. An electrical machine comprising: a rotor; astator adapted to magnetically interact with the rotor, the statorcomprising a plurality of stator teeth, the plurality of stator teethcomprising a first stator tooth comprising a first free end and firstand second channels disposed in the first stator tooth at the first freeend, the first and second channels having a non-rectangular,substantially trapezoidal shape; a second stator tooth comprising asecond free end and third and fourth channels disposed in the secondstator tooth at the second free end, the third and fourth channelshaving the non-rectangular, substantially trapezoidal shape; and a thirdstator tooth comprising a third free end and fifth and sixth channelsdisposed in the third stator tooth at the third free end, the third andfourth channels having the non-rectangular, substantially trapezoidalshape; wherein each of the six channels comprises a respective openinghaving an opening width (w_(b)), a base at a depth from the surface andhaving a base width (w_(n)), a first side at a side angle (α) withrespect to the base, and a second side at a side angle (γ) with respectto the base; wherein the stator defines a first slot opening between thefirst stator tooth and the second stator tooth, and a second slotopening between the second stator tooth and the third stator tooth, eachof the first and second slot openings having a width (w_(o)); andwherein (w_(n)) satisfies the relationship(0.5w_(o))≦(w_(n))≦(1.5w_(o)), (w_(b)) satisfies the relationship(0.3w_(o))≦(w_(b))≦(1.2w_(o)), and (w_(n)) does not equal (w_(b)). 16.An electrical machine as set forth in claim 15 wherein the plurality ofstator teeth further comprises a fourth stator tooth, and wherein thefourth stator tooth comprises a free end and two non-rectangular,substantially trapezoidal channels disposed in the free end.
 17. Anelectrical machine as set forth in claim 15 wherein the electricalmachine comprises a brushless, permanent magnet motor comprising therotor and stator.
 18. An electrical machine comprising: a rotor; astator adapted to magnetically interact with the rotor, the statorcomprising a plurality of stator teeth, the plurality of stator teethcomprising a first stator tooth comprising a first free end and firstand second channels disposed in the first stator tooth at the first freeend, the first and second channels having a non-rectangular,substantially trapezoidal shape; a second stator tooth comprising asecond free end and third and fourth channels disposed in the secondstator tooth at the second free end, the third and fourth channelshaving the non-rectangular, substantially trapezoidal shape; and a thirdstator tooth comprising a third free end and fifth and sixth channelsdisposed in the third stator tooth at the third free end, the third andfourth channels having the non-rectangular, substantially trapezoidalshape; wherein each of the six channels comprises a respective opening,a base at a depth from the surface, a first side at a side angle (α)with respect to the base, and a second side at a side angle (γ) withrespect to the base; and wherein (α) satisfies the relationship(30°)≦(α)<(90°) and wherein (γ) satisfies the relationship(30°)≦(γ)<(90°).
 19. An electrical machine as set forth in claim 18wherein the plurality of stator teeth further comprises a fourth statortooth, and wherein the fourth stator tooth comprises a free end and twonon-rectangular, substantially trapezoidal channels disposed in the freeend.
 20. An electrical machine as set forth in claim 18 wherein theelectrical machine comprises a brushless, permanent magnet motorcomprising the rotor and stator.
 21. An electrical machine comprising: arotor; a stator adapted to magnetically interact with the rotor, thestator comprising a plurality of stator teeth, the plurality of statorteeth comprising a first stator tooth comprising a first free end andfirst and second channels disposed in the first stator tooth at thefirst free end, the first and second channels having a non-rectangular,substantially trapezoidal shape; a second stator tooth comprising asecond free end and third and fourth channels disposed in the secondstator tooth at the second free end, the third and fourth channelshaving the non-rectangular, substantially trapezoidal shape; and a thirdstator tooth comprising a third free end and fifth and sixth channelsdisposed in the third stator tooth at the third free end, the third andfourth channels having the non-rectangular, substantially trapezoidalshape; wherein each of the six channels comprises a respective opening,a base at a depth from the surface, a first side at a side angle (α)with respect to the base, and a second side at a side angle (γ) withrespect to the base; and wherein (α) satisfies the relationship(90°)≦(α)<(135°) and wherein (γ) satisfies the relationship(30°)≦(γ)<(90°).
 22. An electrical machine as set forth in claim 21wherein the plurality of stator teeth further comprises a fourth statortooth, and wherein the fourth stator tooth comprises a free end and twonon-rectangular, substantially trapezoidal channels disposed in the freeend.
 23. An electrical machine as set forth in claim 21 wherein theelectrical machine comprises a brushless, permanent magnet motorcomprising the rotor and stator.
 24. An electrical machine comprising: arotor; a stator adapted to magnetically interact with the rotor, thestator comprising a plurality of stator teeth, the plurality of statorteeth comprising a first stator tooth comprising a first free end andfirst and second channels disposed in the first stator tooth at thefirst free end, the first and second channels having a non-rectangular,substantially trapezoidal shape; a second stator tooth comprising asecond free end and third and fourth channels disposed in the secondstator tooth at the second free end, the third and fourth channelshaving the non-rectangular, substantially trapezoidal shape; and a thirdstator tooth comprising a third free end and fifth and sixth channelsdisposed in the third stator tooth at the third free end, the third andfourth channels having the non-rectangular, substantially trapezoidalshape; wherein each of the six channels comprises a respective openinghaving an opening width (w_(b)), a base at a depth from the surface andhaving a base width (w_(n)), a first side at a side angle (α) withrespect to the base, and a second side at a side angle (γ) with respectto the base; and wherein the first slot opening has a first slotcenterline disposed between the first stator tooth and the second statortooth, and the second slot opening has a second slot centerline disposedbetween the second stator tooth and the third stator tooth, wherein eachof the non-rectangular, substantially trapezoidal channels has a centerline dissecting the opening width (w_(b)), and wherein the centerlinesdissecting the third and fourth opening widths divide the intervalbetween the first and second slot centerlines into three equalintervals.
 25. An electrical machine as set forth in claim 24 whereinthe plurality of stator teeth further comprises a fourth stator tooth,and wherein the fourth stator tooth comprises a free end and twonon-rectangular, substantially trapezoidal channels disposed in the freeend.
 26. An electrical machine as set forth in claim 24 wherein theelectrical machine comprises a brushless, permanent magnet motorcomprising the rotor and stator.